This tutorial provides a clear, visual bridge between abstract integration and simple geometry, making the Fundamental Theorem of Calculus intuitive for any learner. It effectively demonstrates that understanding functional change is often as straightforward as calculating the area of a triangle.
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Area Under f'(x) Gives the Change in f(x)Added:
For this question, we have a graph of F prime. We know F of 0 is 5, but we want to find what is F of 1. So, for this one, we could use the fundamental theorem of calculus. And we have the integral from 0 to 1 of F prime of X is equal to F of 1 minus F of 0. So, if we want to find what is F of 1, F of 1 is equal to F of 0 plus the integral from 0 to 1 of F prime of X DX. And now, since we have a graph of F prime, we could just find the area under the curve since we have a straight line segment here.
So, we're just going to find the area of this triangle and we're going across 1 and up 6. So, we have 1/2 of 1 * 6, which gives us 3. So, now we just plug in F of 0 is equal to 5 plus the area of that triangle is 3. So, our answer is going to be 8, choice D.
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